Course teached as: B020977 - CALCOLO DELLE VARIAZIONI E EQUAZIONI ALLE DERIVATE PARZIALI Second Cycle Degree in MATHEMATICS Curriculum APPLICATIVO
Teaching Language
Italian or English (if non-italian students attend the class)
Course Content
Harmonic functions: classical theory. Dirichlet and Neumann problems: existence and uniqueness. Viscosity solutions of elliptic PDEs: comparison principles, existence, stability. Sobolev spaces and weak solutions of elliptic PDEs. Calculus of Variations: semi-classical and direct methods. Applications to geometric properties of solutions of elliptic equations: quasi-convexity; starshape; symmetry in overdetermined problems; curvature of level surfaces; critical points of solutions.
R. Magnanini, Lecture Notes of the Course of
Calculus of Variations and Partial Differential Equations, (2019). Lecture notes written in English by the teacher.
Learning Objectives
The course aims to acquaint the students with the basic and advanced knowledge and techniques that are necessary to understand and study the modern theory of (elliptic and parabolic) partial differential equations, linear and nonlinear, and some related questions in the field of calculus of variations. At the end of the course the student will learn most of the modern techniques in the study of PDEs and CV. One of the goals is to let the students develop the basic technical skills and the critical thinking needed to apply the acquired knowledge to modeling and solving of problems in different settings, not purely mathematical. Special attention will be paid to help the students to develop communication skills necessary for teamwork. The course covers topics and provides learning skills that are important, or strongly suggested, to undertake a research career in the field of partial differential equations.
Prerequisites
BASICS of Calculus, ODEs, Linear Algebra and Geometry, Real and Complex Analysis.
Teaching Methods
Lectures: Presentation of the theory described in the course program, with teacher-student direct interaction, to ensure a full understanding of the subject.
Training sessions: training of the students to solving a selection of problems with particular emphasis on the study of geometric properties of the solutions of PDEs. The training sessions are conducted so as to help the students to apply and communicate the theoretical knowledge, to improve their ability of independent reasoning.
Moodle learning platform: online teacher-student interaction, posting of additional notes and bibliography, supplementary exercise sheets.
Remark: The suggested reading includes supplementary material that may be useful for further personal studies on the subject.
Further information
Even if, for administrative reasons, this class is in the list of choices in Analysis designed for the Indirizzo Applicativo, the class is a good choice also for the students of the Indirizzo Generale, especially for those who are interested in a career in reaserch.
Type of Assessment
Final oral examination: A selection of exercises is proposed. Besides to assess the students' knowledge and comprehension of the material proposed in the course, the exam is designed to evaluate the students' ability both to present the subject matter in an analytical way and to apply the acquired knowledge and techniques to the solutions of the proposed exercises. In the evaluation, special attention is paid the originality and effectiveness of the methods adopted.
Course program
Introduction to the concepts of classical, viscosity and weak solution. Harmonic functions: mean value property; smoothness; maximum principle; Hopf lemma; Harnack inequality; Liouville's theorem; analyticity; compactness of families of harmonic functions; harmonic functions in the plane. Boundary-value problems: the Poisson equation in the whole space; uniqueness theorems for Dirichlet, Neumann and Robin problems; Green's function; existence by Perron's method; methods for unbounded domains. Viscosity solutions: motivations; equivalent definitions; comparison principles for first and second order operators; Ishii's lemma; uniformly elliptic fully non-linear operators; existence by Perron-Ishii's method; viscosity solutions pass to the limit. Introduction to Sobolev spaces: weak derivativatives; definition of Sobolev spaces; approximation and extension; Sobolev inequalities; embeddings; compactness; Poincaré inequalities. Methods in the Calculus of Variations: the Dirichlet principle; the bounded slope condition; existence of non-parametric minimal surfaces; the direct method. Geometric properties of solutions of elliptic PDEs: the Green's, the torsion and the capacity functions; quasi-convexity: Korevaar's method and the method of the quasi-convex envelope; starshape; symmetry in over-determined boundary-value problems: shape-optimization problems, Alexandrov's method of reflection, Serrin's method of moving planes, symmetry by integral identities, Pohozaev's identity, stability of spherical symmetry; curvature of level surfaces; critical points of solutions; hot spots.